AI 中文总结
研究数域\(K\)上非常数有理函数\(f_1,\ldots,f_n\)，在其不能乘法生成线性分式函数幂的条件下，探讨使\(f_1(\alpha),\ldots,f_n(\alpha)\)模特定子集乘法相关的\(\alpha\in K\)的情况，改进了先前结果。
AI 中文摘要
本文表明，若数域\(K\)上的非常数有理函数\(f_1,\ldots,f_n\in K(x)\)不能乘法生成线性分式函数的幂，则只有有限多个\(\alpha\in K\)，使得\(f_1(\alpha),\ldots,f_n(\alpha)\)相对于\(K\)的有限生成乘法子群的除法群（关于韦伊高度）在某个‘接近’的子集中是乘法相关的。这改进了一些先前的结果。
英文摘要
In this paper, we show that if the non-constant rational functions $f_1, \ldots, f_n\in K(x)$ over a number field $K$ cannot multiplicatively generate a power of a linear fractional function, then there are only finitely many elements $α\in K$ such that $f_1(α),\ldots,f_n(α)$ are multiplicatively dependent modulo some subset `close' (with respect to the Weil height) to the division group of a finitely generated multiplicative subgroup of $K$. This improves some previous results.
